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Econ-56 · economics · Economics

Minimax estimation of longitudinal intervention effects

LLM status Open unverified

Statement

CI-47 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Observe $n$ independent trajectories $O=(X,A_0,L,A_1,Y)$, where $X\in[0,1]^{d_0}$, $L\in[0,1]^{d_1}$, and $A_0,A_1,Y$ are binary. Fix a treatment regime $a=(a_0,a_1)$. Assume consistency, sequential exchangeability given the observed history at each treatment time, and propensities $e_0(x)$ and $e_1(x,a_0,l)$ between $\varepsilon$ and $1-\varepsilon$. Let \[ \psi_a(P)=\int m_a(x,l)g_{a_0}(l\mid x)p(x)\,dl\,dx, \] where $m_a(x,l)=\E[Y\mid X=x,A_0=a_0,L=l,A_1=a_1]$. This is $\E[Y(a_0,a_1)]$ under the stated causal assumptions. For fixed dimensions and constants, place $p,e_0$ in Hölder balls of dimension $d_0$, and each treatment-specific $g_{a_0},e_1,m_a$ in balls of dimension $d_0+d_1$, with specified positive smoothness indices $s_p,s_0,s_g,s_1,s_m$. Require $p$ and the transition densities to be bounded above and away from zero.

Formal question

Determine matching upper and lower bounds for \[ \inf_{\widehat\psi_a}\sup_{P\in\mathcal P(s_p,s_0,s_g,s_1,s_m)} \E_P[(\widehat\psi_a-\psi_a(P))^2]. \] Identify the $n^{-1}$ mean-squared-error region and the slower regions, specifying which interaction among the two propensity functions, transition density, and final outcome regression determines each rate. Extend the characterization to a fixed number of treatment times with explicitly stated history dimensions.

Scope and status

Sequential exchangeability is an identifying assumption, not a conclusion from observed trajectories. Two treatment decisions already make the minimax experiment precise; allowing an unspecified growing horizon would define a different question. These model choices formalize an agenda and do not assert that every special case remains unsolved.

References for the source of the problem

Carlos Cinelli, Avi Feller, Guido Imbens, Edward Kennedy, Sara Magliacane, and Jose Zubizarreta, Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1, 23 August 2025, Section 3.5.3, New parameters. https://arxiv.org/html/2508.17099v1#S3.SS5.SSS3

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 3

Econ-56-A3 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-56, version 3

Minimax estimation of longitudinal intervention effects: Doubly robust nested regression and a larger parametric region

Problem ID: C14-55. Status: Unresolved; rigorous restricted results.

Target

For a fixed two-stage regime $a=(a_0,a_1)$, write $e_0(x)=P(A_0=a_0\mid X=x)$, $e_1(x,l)=P(A_1=a_1\mid X=x,A_0=a_0,L=l)$, $g(l\mid x)=p(l\mid x,A_0=a_0)$, and $m(x,l)=\mathbb E[Y\mid x,a_0,l,a_1]$. Then \[ Q(x)=\int m(x,l)g(l\mid x)dl,\qquad \psi_a=\int Q(x)p(x)dx. \] The canonical question seeks the squared-error minimax rate under distinct Hölder restrictions on all five nuisance functions. The agenda is [agenda, Section 3.5.3]. Longitudinal higher-order methodology already exists, including [robins]; this note supplies an exact first-order remainder, a constructive direct-regression upper bound with a matching parametric subregion, and a sharp known-assignment experiment. The full longitudinal smoothness phase diagram remains unresolved.

A count identity used in the regression constructions

Lemma: If $J\sim\operatorname{Bin}(N,\rho)$ with $N\ge0$ and $0<\rho\le1$, then \[ \mathbb E\frac1{J+1} =\frac{1-(1-\rho)^{N+1}}{(N+1)\rho},\qquad \mathbb E\frac{\mathbf1\{J>0\}}J \le\frac2{(N+1)\rho},\qquad \Pr(J=0)\le\frac1{(N+1)\rho}. \]
Proof: Integrate $\mathbb E t^J=(1-\rho+\rho t)^N$ from zero to one. The finite sum and integral can be interchanged, giving the first identity. For $J>0$, $1/J\le2/(J+1)$, while $\mathbf1\{J=0\}\le1/(J+1)$ for every $J$. Taking expectations and bounding the first numerator by one proves both inequalities. ∎

Exact two-stage bias

Train pilots $h_0,h_1,\widehat m,\widehat Q$ independently of an evaluation block of size $N$, with $h_t\in[\varepsilon,1-\varepsilon]$ and outcome pilots in $[0,1]$. Put $I_t=\mathbf1\{A_t=a_t\}$ and average \[ S=\widehat Q(X)+\frac{I_0}{h_0(X)} \{\widehat m(X,L)-\widehat Q(X)\} +\frac{I_0I_1}{h_0(X)h_1(X,L)}\{Y-\widehat m(X,L)\}. \tag{1} \] Let $\mu(dx,dl)=p(x)g(l\mid x)dx\,dl$.
Theorem: Conditionally on the pilots, the bias of (1) is \[ B=\int p\left(1-\frac{e_0}{h_0}\right)(\widehat Q-Q)dx +\int\frac{e_0}{h_0}\left(1-\frac{e_1}{h_1}\right) (\widehat m-m)d\mu. \tag{2} \] Consequently its projected mean has risk bounded by \[ \frac{C_\varepsilon}{N}+C_\varepsilon\mathbb E\left[ \|h_0-e_0\|_{L^2(p)}^2\|\widehat Q-Q\|_{L^2(p)}^2 +\|h_1-e_1\|_{L^2(\mu)}^2\|\widehat m-m\|_{L^2(\mu)}^2\right]. \tag{3} \]
Proof: Conditional on $X=x$, the expectation of the second and third terms in (1) equals \[ \frac{e_0}{h_0}\int g\left\{\widehat m-\widehat Q +\frac{e_1}{h_1}(m-\widehat m)\right\}dl =\frac{e_0}{h_0}\left\{Q-\widehat Q+ \int g(1-e_1/h_1)(\widehat m-m)dl\right\}. \] Adding $\widehat Q$ proves (2). Cauchy--Schwarz and the clipped denominator bounds give the two products in (3), after $(x+y)^2\le2x^2+2y^2$. The score is bounded by $1+\varepsilon^{-1}+\varepsilon^{-2}$, so its conditional variance divided by $N$ proves the remaining term. ∎
Proposition: If $g\le G$, $\widehat g$ is a nonnegative conditional density, and $\widehat Q(x)=\int\widehat m(x,l)\widehat g(l\mid x)dl$, then \[ \|\widehat Q-Q\|_{L^2(p)} \le \|\widehat m-m\|_{L^2(\mu)} +\|\widehat g-g\|_{L^2(p(x)dx\,dl)}. \tag{4} \] In particular, suppose the evaluation size satisfies $N\asymp n$, and all four pilot errors have fourth moments of orders $n^{-4\gamma_0},n^{-4\gamma_1},n^{-4\gamma_m},n^{-4\gamma_g}$ in these respective norms. Then (3) is at most \[ C\{n^{-1}+n^{-2(\gamma_0+\min(\gamma_m,\gamma_g))} +n^{-2(\gamma_1+\gamma_m)}\}. \tag{5} \]
Proof: Write $\widehat Q-Q=\int(\widehat m-m)g+ \int\widehat m(\widehat g-g)$. Jensen bounds the first term's norm by the first term on the right of (4). Cauchy--Schwarz in $l$, whose unit cube has volume one, and $0\le\widehat m\le1$ bound the second term. Minkowski proves (4). For (5), apply Cauchy--Schwarz to the expectations of products in (3), then (4) and $(x+y)^4\le8(x^4+y^4)$. ∎
Thus an explicitly sufficient parametric region for any pilots satisfying these moment bounds is $\gamma_0+\min(\gamma_m,\gamma_g)\ge1/2$ and $\gamma_1+\gamma_m\ge1/2$. This is a conditional statement about pilot performance, not an unproved assertion that every indicated exponent is attainable for arbitrary joint Hölder indices. Formula (2) also shows that density and transition errors do not necessarily enter as independent additive first-order errors. In particular, a sufficiently good direct $Q$ estimator may be preferable to separately estimating $g$.

An exact result with supplied assignment probabilities

Consider $T$ treatment decisions, each with the probability of the requested action at least $\varepsilon$. Histories and their transitions may otherwise be arbitrary. Let $p_t$ denote those supplied, history-dependent probabilities, $Y\in[0,1]$, and retain sequential exchangeability and consistency.
Theorem: The minimax risk over this known-assignment experiment is at most \[ \min\{1,\,(n\varepsilon^T)^{-1}\}. \] Over the subexperiment with independent assignments of probability $\varepsilon$, ancillary uniform continuous histories, and an unknown constant Bernoulli outcome mean only on the requested treatment path, it is at least $c\min\{1,(n\varepsilon^T)^{-1}\}$ for a universal $c>0$. All continuous-coordinate transition and outcome functions in this subexperiment are smooth of every order.
Proof: Let $R=\prod_{t=1}^T\mathbf1\{A_t=a_t\}/p_t(H_t)$. Iterated conditional expectations, or cancellation of the assignment factors in the joint density, give $\mathbb E[RY]=\psi_a$ and $\mathbb E R=1$. Since $R\le\varepsilon^{-T}$, $\mathbb E(RY)^2\le\mathbb E R^2\le\varepsilon^{-T}$. The projected empirical mean therefore has risk at most $(n\varepsilon^T)^{-1}$; the constant estimator gives the other bound. For the lower bound let $q=\varepsilon^T$, set all off-path outcome means to $1/2$, and set the on-path mean to $1/2\pm\Delta$. The one-observation KL is at most $Cq\Delta^2$ for $0<\Delta\le1/4$. Choose $\Delta=b\min\{1,(nq)^{-1/2}\}$ with $b$ small. Then sample total variation is at most $1/2$ and the two targets differ by $2\Delta$. Testing by the nearer target gives maximal squared risk at least $\Delta^2/4$. Uniform ancillary histories do not change this likelihood calculation or any smoothness constraint with sufficient fixed radius. ∎

A direct nested regression with explicit attainable rates

This section constructs all pilots rather than assuming their moment rates. It uses the canonical five-function model and the fixed two-stage horizon. Let $D=d_0+d_1$, and set \[ t_0=\min(s_0,1),\quad t_1=\min(s_1,1),\quad t_m=\min(s_m,1),\quad t_Q=\min(s_m,s_g,1), \] \[ \gamma_0=\frac{t_0}{2t_0+d_0},\quad \gamma_1=\frac{t_1}{2t_1+D},\quad \gamma_m=\frac{t_m}{2t_m+D},\quad \gamma_Q=\frac{t_Q}{2t_Q+d_0}. \tag{6} \] All these indices are supplied for this minimax upper bound. We make no adaptation claim here. The construction estimates $Q$ directly from a pseudo-outcome and never estimates $g$ or $p$.
Lemma (Integrated regression and contamination): The integrated regression $Q$ has a Hölder modulus of order $t_Q$, with a constant depending only on the fixed dimensions and radii. Let $\widetilde m:[0,1]^D\to[0,1]$ be any random function independent of a new block of $N$ trajectories. On that block regress the response $\widetilde m(X,L)$ among observations with $A_0=a_0$ on $X$ by a histogram of side length $h$, using $1/2$ in empty cells. Call the result $\widetilde Q$. Then, conditionally on $\widetilde m$, \[ \mathbb E[\|\widetilde Q-Q\|_{L^2(p)}^2\mid\widetilde m] \le C\left\{h^{2t_Q}+\frac{h^{-d_0}}{N+1} +\|\widetilde m-m\|_{L^2(\mu)}^2\right\}. \tag{7} \] No smoothness of the fitted function $\widetilde m$ is required.
Proof: For two covariate points, add and subtract $m(x',l)g(l\mid x)$: \[ Q(x)-Q(x')=\int[m(x,l)-m(x',l)]g(l\mid x)dl +\int m(x',l)[g(l\mid x)-g(l\mid x')]dl. \] The first term is bounded by a constant times $\|x-x'\|^{\min(s_m,1)}$, since $g$ integrates to one. The second is bounded by a constant times $\|x-x'\|^{\min(s_g,1)}$, since $0\le m\le1$ and the $l$-cube has volume one. Indices above one imply Lipschitz moduli through bounded first derivatives. The bounded support absorbs the constants needed to replace both exponents by $t_Q$.

For an $X$-cell $A$, write $\pi_A=\int_Ap$ and $\rho_A=\int_A e_0p\ge\varepsilon\pi_A$. The true selected mean of $Q$ is $q_A=\rho_A^{-1}\int_A e_0p Q$ and differs from every $Q(x)$ in that cell by at most $Ch^{t_Q}$. Conditional on $\widetilde m$, the selected pseudo-response mean in the cell is $q_A+r_A$, where \[ r_A=\frac1{\rho_A}\int_A e_0(x)p(x) \int [\widetilde m(x,l)-m(x,l)]g(l\mid x)\,dl\,dx. \] Jensen's inequality for the selected conditional law gives \[ \sum_A\pi_A r_A^2 \le\sum_A\frac{\pi_A}{\rho_A}\int_Ae_0p \int(\widetilde m-m)^2g\,dl\,dx \le\varepsilon^{-1}\|\widetilde m-m\|_{L^2(\mu)}^2. \tag{8} \] On a positive count $j$, the empirical pseudo-response mean is unbiased for $q_A+r_A$ and has conditional variance at most $1/(4j)$. The binomial reciprocal identity and empty-cell bound used earlier give an integrated sampling contribution at most $C h^{-d_0}/(N+1)$. Using $(u+v)^2\le2u^2+2v^2$ for approximation and contamination, then (8), proves (7). The argument conditioned on the whole random function first; it therefore does not assume that a fitted histogram is smooth or that its evaluation errors are independent across cells. ∎

Theorem (Constructive five-index upper bound): There is an explicitly tuned, five-block estimator with \[ \sup_{P\in\mathcal P(s_p,s_0,s_g,s_1,s_m)} \mathbb E_P(\widehat\psi_a-\psi_a)^2 \le C\left\{n^{-1}+n^{-2(\gamma_0+\gamma_Q)} +n^{-2(\gamma_0+\gamma_m)}+n^{-2(\gamma_1+\gamma_m)}\right\}. \tag{9} \] Its constant depends only on the fixed dimensions, indices, radii, density bounds and overlap. Thus \[ \gamma_0+\min(\gamma_Q,\gamma_m)\ge\tfrac12,\qquad \gamma_1+\gamma_m\ge\tfrac12 \tag{10} \] is an attained parametric-risk region. If the class contains a fixed nondegenerate constant Bernoulli outcome subfamily with uniform histories and constant assignments, its minimax risk is $\Theta(n^{-1})$ throughout (10).
Proof: For sufficiently large $n$, use five disjoint blocks of size $N=\lfloor n/5\rfloor$. On block 1 fit $h_0$ by regressing $I_0=\mathbf1\{A_0=a_0\}$ on $X$. On block 2 fit $h_1$ by regressing $I_1=\mathbf1\{A_1=a_1\}$ on $(X,L)$ among $I_0=1$. On block 3 fit $\widehat m$ by regressing $Y$ on $(X,L)$ among $I_0I_1=1$. Use equal-cube histograms with side lengths within a fixed factor of $N^{-1/(2t_0+d_0)}$, $N^{-1/(2t_1+D)}$, and $N^{-1/(2t_m+D)}$, respectively. Clip the propensity fits to $[\varepsilon,1-\varepsilon]$ and the outcome fit to $[0,1]$.

Here is the elementary risk calculation for these pilots. For each cell the regression oscillation is at most a constant times its side length to the relevant $t$ power. The positive-count response mean is unbiased for its selected cell mean and has variance at most $1/(4j)$. For the $h_1$ fit the selected cell probability is $\int_{\text{cell}}e_0\,d\mu\ge\varepsilon\mu(\text{cell})$; for the $\widehat m$ fit it is $\int_{\text{cell}}e_0e_1\,d\mu\ge\varepsilon^2\mu(\text{cell})$. For $h_0$ it is $P_X(\text{cell})$. Multiplying the binomial reciprocal and empty-cell bounds by the respective target cell masses and summing gives risk at most $C\{h^{2t}+h^{-d'}/(N+1)\}$ in dimension $d'$. This argument uses the intervention design measure $\mu=p g$ for the last two errors; it does not silently replace it by the observed marginal law of $(X,L)$. The resulting bounds are \[ \mathbb E\|h_0-e_0\|_{L^2(p)}^2\le Cn^{-2\gamma_0},\quad \mathbb E\|h_1-e_1\|_{L^2(\mu)}^2\le Cn^{-2\gamma_1},\quad \mathbb E\|\widehat m-m\|_{L^2(\mu)}^2\le Cn^{-2\gamma_m}. \tag{11} \]

On block 4 construct $\widehat Q$ from pseudo-response $\widehat m(X,L)$ as in (7), using side length $h_Q\asymp N^{-1/(2t_Q+d_0)}$. Independence of blocks 3 and 4, followed by (7) and (11), gives \[ \mathbb E\|\widehat Q-Q\|_{L^2(p)}^2 \le C\{n^{-2\gamma_Q}+n^{-2\gamma_m}\}. \tag{12} \] The fit already lies in $[0,1]$. On block 5 average the longitudinal score (1) and project its average to $[0,1]$.

The exact remainder (2) and bounded-score variance give (3). The function $h_0$ uses only block 1, whereas $\widehat Q$ uses only blocks 3 and 4. Thus their squared errors in the first product in (3) are independent. Similarly $h_1$ uses only block 2 and $\widehat m$ only block 3, so the second product also factors in expectation. Independence of $\widehat Q$ and $\widehat m$ is neither true nor needed. Substitution of (11)--(12) into (3) proves (9) using only second-moment pilot bounds, without the fourth-moment premise of (5). Small sample sizes are handled by the constant estimator and a larger constant. Condition (10) makes all displayed terms $O(n^{-1})$.

For the matching lower bound in that region, let histories be independent uniform variables, assignments independent Bernoulli$(1/2)$, and $Y\mid X,A_0,L,A_1\sim\operatorname{Ber}(\theta)$ with $\theta=1/2\pm b/\sqrt n$. The target is $\theta$ and the outcome law is constant in every continuous coordinate, hence lies in every stated smoothness ball under the explicit subfamily assumption. Product KL is $O(b^2)$. Choose fixed $b>0$ so total variation is at most $1/2$; the nearer-target test yields a lower bound $c/n$ for squared-error risk. Together with (9) this proves the restricted sharp rate. ∎

For a concrete improvement, take $d_0=d_1=1$, $s_0=s_1=s_m=1$ and $s_g=2/5$, with any positive $s_p$. Then \[ \gamma_0=1/3,\qquad \gamma_1=\gamma_m=1/4,\qquad \gamma_Q=2/9, \] so (10) holds. Separate histogram estimation of the transition density would instead yield \[ \gamma_g=\frac{2/5}{2(2/5)+2}=\frac17,\qquad \gamma_0+\min(\gamma_m,\gamma_g)=\frac{10}{21}<\frac12, \] which does not satisfy the earlier sufficient condition. The comparison concerns these proved first-order constructions; it is not a lower bound against every estimator that explicitly estimates $g$.

Version 3: regress the doubly robust pseudo-outcome

Version 2 regressed $\widehat m(X,L)$ to construct the first-stage regression. Its contamination term in (12) has the rate of $\widehat m$ alone. We now replace that response by a second-stage augmented response. This changes the proved sufficient region, while preserving the same canonical two-stage model and the capped histogram approximation orders. The indices $s_p,s_0,s_g,s_1,s_m$ are supplied for this minimax construction; no adaptation to unknown indices is asserted.

Retain $t_j=\min(s_j,1)$, $D=d_0+d_1$, $t_Q=\min(s_m,s_g,1)$ and \[ \gamma_0=\frac{t_0}{2t_0+d_0},\quad \gamma_1=\frac{t_1}{2t_1+D},\quad \gamma_m=\frac{t_m}{2t_m+D},\quad \gamma_Q=\frac{t_Q}{2t_Q+d_0}. \] The density assumptions used below are the canonical bounds $0<c_p\le p\le C_p$ and $0<c_g\le g\le C_g$ on their full cubes. They yield a lower bound for the selected design density, not merely an integrated overlap condition. All constants below may depend on these bounds, fixed dimensions, smoothness indices, radii and $\varepsilon$.

Lemma (Pointwise histogram error and a second-order contamination): Construct $h_1$ and $\widehat m$ on independent blocks of $N$ trajectories, exactly as above, with side lengths tuned to $t_1,t_m$. Then, for almost every $z=(x,l)$, uniformly in $z$ and the model, \[ \mathbb E|h_1(z)-e_1(z)|^2\le C N^{-2\gamma_1},\qquad \mathbb E|\widehat m(z)-m(z)|^2\le C N^{-2\gamma_m}. \tag{13} \] On a third, independent block define \[ U=\widehat m(X,L)+\frac{I_1}{h_1(X,L)} \{Y-\widehat m(X,L)\}. \] Conditionally on both fitted functions, its selected regression is \[ \begin{aligned} \mathbb E[U\mid X=x,I_0=1,h_1,\widehat m]&=Q(x)+r(x),\\ r(x)&=\int g(l\mid x) \left(1-\frac{e_1(x,l)}{h_1(x,l)}\right) [\widehat m(x,l)-m(x,l)]\,dl. \end{aligned} \tag{14} \] Consequently \[ \mathbb E\|r\|_{L^2(p)}^2 \le C N^{-2(\gamma_1+\gamma_m)}. \tag{15} \]
Proof: For a histogram cell of side $h$ in dimension $D$, the probability of selection and membership is at least $c_p c_g\varepsilon h^D$ for $h_1$, and at least $c_p c_g\varepsilon^2h^D$ for $\widehat m$. The selected cell mean differs from the target regression at every point of the cell by at most $C h^t$. Given a positive count $j$, the sample response mean has variance at most $1/(4j)$. The binomial reciprocal identity already proved, including the empty-cell default, therefore bounds pointwise squared error by $C\{h^{2t}+[(N+1)h^D]^{-1}\}$. Clipping cannot increase the error. Substitution of the two bandwidths proves (13); importantly, no integration over $z$ was required for this bound.

Conditional on $X,L,I_0=1$, the expectation of $I_1Y$ is $e_1m$. Expanding $U$ and integrating over $L$ gives (14). Jensen's inequality for the conditional density $g$, and $h_1\ge\varepsilon$, give \[ \|r\|_{L^2(p)}^2\le\varepsilon^{-2} \int (h_1-e_1)^2(\widehat m-m)^2\,d\mu. \] At every fixed $z$, the two squared errors are independent because their entire training blocks are independent. Taking expectations, using Tonelli's theorem for the nonnegative integrand, and applying the two uniform bounds in (13) proves (15). This factorization does not assert independence of prediction errors at different covariate points. ∎

On the independent $U$ block, among observations with $I_0=1$, regress $U$ on $X$ by an equal-cell histogram with $h_Q\asymp N^{-1/(2t_Q+d_0)}$, use default $1/2$ for empty cells, and clip the fitted function to $[0,1]$. Denote it by $\widehat Q_{\rm dr}$. The response need not lie in $[0,1]$, but it obeys $|U|\le 1+1/\varepsilon$ deterministically.

Lemma (Risk of the augmented nested fit): For this construction, \[ \mathbb E\|\widehat Q_{\rm dr}-Q\|_{L^2(p)}^2 \le C\{N^{-2\gamma_Q}+N^{-2(\gamma_1+\gamma_m)}\}. \tag{16} \]
Proof: Condition on the two pilots. In an $X$-cell $A$, let $\rho_A=\int_A e_0p$ and $\pi_A=\int_A p$. The conditional mean of a selected response equals the selected cell average of $Q+r$. The selected average of $Q$ differs from $Q(x)$ throughout the cell by at most $C h_Q^{t_Q}$, by the integrated regression modulus proved earlier. The selected average $r_A$ of $r$ satisfies \[ \sum_A\pi_A r_A^2 \le\sum_A\frac{\pi_A}{\rho_A}\int_A e_0p r^2 \le\varepsilon^{-1}\|r\|_{L^2(p)}^2. \] A positive-count response mean has conditional variance at most $(1+1/\varepsilon)^2/j$. The same count calculation as in (7) gives an integrated variance and empty-cell contribution $C h_Q^{-d_0}/(N+1)$. Use $(a+b+c)^2\le3(a^2+b^2+c^2)$ for sampling error, approximation and contamination; clipping reduces error to the true $Q\in[0,1]$. Thus the conditional risk is at most $C\{h_Q^{2t_Q}+h_Q^{-d_0}/(N+1)+\|r\|_2^2\}$. Average over the pilots, use (15), and tune $h_Q$ to prove (16). ∎
Theorem (A strictly enlarged attained parametric region): There is a five-block estimator, using the augmented fit above in the longitudinal score (1), such that \[ \sup_{P\in\mathcal P(s_p,s_0,s_g,s_1,s_m)} \mathbb E_P(\widehat\psi_a-\psi_a)^2 \le C\left\{n^{-1}+n^{-2(\gamma_0+\gamma_Q)} +n^{-2(\gamma_1+\gamma_m)}\right\}. \tag{17} \] Under the same constant-subfamily condition used for the lower bound above, the squared-error minimax risk is $\Theta(n^{-1})$ whenever \[ \gamma_0+\gamma_Q\ge\tfrac12,\qquad \gamma_1+\gamma_m\ge\tfrac12. \tag{18} \]
Proof: Use blocks 1, 2 and 3 to fit $h_0,h_1,\widehat m$ respectively; block 4 to fit $\widehat Q_{\rm dr}$ from $U$; and block 5 to average (1) with $\widehat Q=\widehat Q_{\rm dr}$ and project its mean to $[0,1]$. All block sizes are $N=\lfloor n/5\rfloor$; small $n$ are handled by a constant estimator. The exact remainder (2) is valid for this new fit without alteration. The first product in (3) factors in expectation because $h_0$ uses only block 1 and $\widehat Q_{\rm dr}$ uses only blocks 2--4. The second product factors because $h_1$ and $\widehat m$ use independent blocks 2 and 3. Their mutual dependence with $\widehat Q_{\rm dr}$ is immaterial to this sum of two bounds. Using (11) and (16) gives the right side of (17), plus $C n^{-2(\gamma_0+\gamma_1+\gamma_m)}$. Since $\gamma_0>0$, this extra term is bounded by the last term in (17). The bounded-score evaluation variance is $C/n$. This proves (17). Conditions (18) give the upper parametric rate; the constant Bernoulli two-point lower bound previously proved supplies the matching lower rate in every such class. ∎

For a strict comparison take $d_0=d_1=1$, $s_0=1/3$ and $s_1=s_m=s_g=1$, with any positive $s_p$. Then $\gamma_0=1/5$, $\gamma_1=\gamma_m=1/4$ and $\gamma_Q=1/3$. The new sums are $8/15$ and $1/2$, so (18) holds, whereas version 2 additionally required $\gamma_0+\gamma_m=9/20\ge1/2$, which fails. This is a strict enlargement of the region established by these constructions; it is not an impossibility theorem for other estimators outside that region. Doubly robust augmentation is a standard methodological tool [bang]; the new contribution within this attempt is the fully proved, index-specific improvement over its previous bound.

Unresolved minimax interactions

The augmented nested fit removes the extra first-stage-propensity times final-outcome-regression condition from the previous sufficient region. The result retains the canonical unknown transition and assignment laws, but its histogram approximation orders remain capped at one. Conditions (18) are sufficient, not claimed necessary. Higher-order corrections, matching slow-region lower bounds, the full five-index phase diagram and the fixed-multistage characterization remain unresolved here. The earlier known-assignment horizon obstruction concerns a separate growing-horizon experiment and does not supply those missing lower bounds.

Completion Estimate

40%

This is a heuristic estimate for the full catalogue target. Version 3 gives a strict, constructive enlargement of the attained parametric region by proving the second-order contamination bound and all required sample independences. The full sharp slow-rate characterization and its fixed-multistage extension remain missing.

References

  • [agenda] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2025), arXiv:2508.17099v1. https://arxiv.org/html/2508.17099v1.
  • [robins] J. Robins, L. Li, E. Tchetgen Tchetgen and A. van der Vaart. Technical Report: Higher Order Influence Functions and Minimax Estimation of Nonlinear Functionals (2016). https://arxiv.org/abs/1601.05820.
  • [bang] H. Bang and J. M. Robins. Doubly Robust Estimation in Missing Data and Causal Inference Models. Biometrics 61 (2005), 962--973.

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Econ-56-A2 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-56, version 2

Minimax estimation of longitudinal intervention effects: Direct nested regression and an attained five-index parametric region

Problem ID: C14-55. Status: Unresolved; rigorous restricted results.

Target

For a fixed two-stage regime $a=(a_0,a_1)$, write $e_0(x)=P(A_0=a_0\mid X=x)$, $e_1(x,l)=P(A_1=a_1\mid X=x,A_0=a_0,L=l)$, $g(l\mid x)=p(l\mid x,A_0=a_0)$, and $m(x,l)=\mathbb E[Y\mid x,a_0,l,a_1]$. Then \[ Q(x)=\int m(x,l)g(l\mid x)dl,\qquad \psi_a=\int Q(x)p(x)dx. \] The canonical question seeks the squared-error minimax rate under distinct Hölder restrictions on all five nuisance functions. The agenda is [agenda, Section 3.5.3]. Longitudinal higher-order methodology already exists, including [robins]; this note supplies an exact first-order remainder, a constructive direct-regression upper bound with a matching parametric subregion, and a sharp known-assignment experiment. The full longitudinal smoothness phase diagram remains unresolved.

Exact two-stage bias

Train pilots $h_0,h_1,\widehat m,\widehat Q$ independently of an evaluation block of size $N$, with $h_t\in[\varepsilon,1-\varepsilon]$ and outcome pilots in $[0,1]$. Put $I_t=\mathbf1\{A_t=a_t\}$ and average \[ S=\widehat Q(X)+\frac{I_0}{h_0(X)} \{\widehat m(X,L)-\widehat Q(X)\} +\frac{I_0I_1}{h_0(X)h_1(X,L)}\{Y-\widehat m(X,L)\}. \tag{1} \] Let $\mu(dx,dl)=p(x)g(l\mid x)dx\,dl$.
Theorem: Conditionally on the pilots, the bias of (1) is \[ B=\int p\left(1-\frac{e_0}{h_0}\right)(\widehat Q-Q)dx +\int\frac{e_0}{h_0}\left(1-\frac{e_1}{h_1}\right) (\widehat m-m)d\mu. \tag{2} \] Consequently its projected mean has risk bounded by \[ \frac{C_\varepsilon}{N}+C_\varepsilon\mathbb E\left[ \|h_0-e_0\|_{L^2(p)}^2\|\widehat Q-Q\|_{L^2(p)}^2 +\|h_1-e_1\|_{L^2(\mu)}^2\|\widehat m-m\|_{L^2(\mu)}^2\right]. \tag{3} \]
Proof: Conditional on $X=x$, the expectation of the second and third terms in (1) equals \[ \frac{e_0}{h_0}\int g\left\{\widehat m-\widehat Q +\frac{e_1}{h_1}(m-\widehat m)\right\}dl =\frac{e_0}{h_0}\left\{Q-\widehat Q+ \int g(1-e_1/h_1)(\widehat m-m)dl\right\}. \] Adding $\widehat Q$ proves (2). Cauchy--Schwarz and the clipped denominator bounds give the two products in (3), after $(x+y)^2\le2x^2+2y^2$. The score is bounded by $1+\varepsilon^{-1}+\varepsilon^{-2}$, so its conditional variance divided by $N$ proves the remaining term. ∎
Proposition: If $g\le G$, $\widehat g$ is a nonnegative conditional density, and $\widehat Q(x)=\int\widehat m(x,l)\widehat g(l\mid x)dl$, then \[ \|\widehat Q-Q\|_{L^2(p)} \le \|\widehat m-m\|_{L^2(\mu)} +\|\widehat g-g\|_{L^2(p(x)dx\,dl)}. \tag{4} \] In particular, suppose the evaluation size satisfies $N\asymp n$, and all four pilot errors have fourth moments of orders $n^{-4\gamma_0},n^{-4\gamma_1},n^{-4\gamma_m},n^{-4\gamma_g}$ in these respective norms. Then (3) is at most \[ C\{n^{-1}+n^{-2(\gamma_0+\min(\gamma_m,\gamma_g))} +n^{-2(\gamma_1+\gamma_m)}\}. \tag{5} \]
Proof: Write $\widehat Q-Q=\int(\widehat m-m)g+ \int\widehat m(\widehat g-g)$. Jensen bounds the first term's norm by the first term on the right of (4). Cauchy--Schwarz in $l$, whose unit cube has volume one, and $0\le\widehat m\le1$ bound the second term. Minkowski proves (4). For (5), apply Cauchy--Schwarz to the expectations of products in (3), then (4) and $(x+y)^4\le8(x^4+y^4)$. ∎
Thus an explicitly sufficient parametric region for any pilots satisfying these moment bounds is $\gamma_0+\min(\gamma_m,\gamma_g)\ge1/2$ and $\gamma_1+\gamma_m\ge1/2$. This is a conditional statement about pilot performance, not an unproved assertion that every indicated exponent is attainable for arbitrary joint Hölder indices. Formula (2) also shows that density and transition errors do not necessarily enter as independent additive first-order errors. In particular, a sufficiently good direct $Q$ estimator may be preferable to separately estimating $g$.

An exact result with supplied assignment probabilities

Consider $T$ treatment decisions, each with the probability of the requested action at least $\varepsilon$. Histories and their transitions may otherwise be arbitrary. Let $p_t$ denote those supplied, history-dependent probabilities, $Y\in[0,1]$, and retain sequential exchangeability and consistency.
Theorem: The minimax risk over this known-assignment experiment is at most \[ \min\{1,\,(n\varepsilon^T)^{-1}\}. \] Over the subexperiment with independent assignments of probability $\varepsilon$, ancillary uniform continuous histories, and an unknown constant Bernoulli outcome mean only on the requested treatment path, it is at least $c\min\{1,(n\varepsilon^T)^{-1}\}$ for a universal $c>0$. All continuous-coordinate transition and outcome functions in this subexperiment are smooth of every order.
Proof: Let $R=\prod_{t=1}^T\mathbf1\{A_t=a_t\}/p_t(H_t)$. Iterated conditional expectations, or cancellation of the assignment factors in the joint density, give $\mathbb E[RY]=\psi_a$ and $\mathbb E R=1$. Since $R\le\varepsilon^{-T}$, $\mathbb E(RY)^2\le\mathbb E R^2\le\varepsilon^{-T}$. The projected empirical mean therefore has risk at most $(n\varepsilon^T)^{-1}$; the constant estimator gives the other bound. For the lower bound let $q=\varepsilon^T$, set all off-path outcome means to $1/2$, and set the on-path mean to $1/2\pm\Delta$. The one-observation KL is at most $Cq\Delta^2$ for $0<\Delta\le1/4$. Choose $\Delta=b\min\{1,(nq)^{-1/2}\}$ with $b$ small. Then sample total variation is at most $1/2$ and the two targets differ by $2\Delta$. Testing by the nearer target gives maximal squared risk at least $\Delta^2/4$. Uniform ancillary histories do not change this likelihood calculation or any smoothness constraint with sufficient fixed radius. ∎

A direct nested regression with explicit attainable rates

This section constructs all pilots rather than assuming their moment rates. It uses the canonical five-function model and the fixed two-stage horizon. Let $D=d_0+d_1$, and set \[ t_0=\min(s_0,1),\quad t_1=\min(s_1,1),\quad t_m=\min(s_m,1),\quad t_Q=\min(s_m,s_g,1), \] \[ \gamma_0=\frac{t_0}{2t_0+d_0},\quad \gamma_1=\frac{t_1}{2t_1+D},\quad \gamma_m=\frac{t_m}{2t_m+D},\quad \gamma_Q=\frac{t_Q}{2t_Q+d_0}. \tag{6} \] All these indices are supplied for this minimax upper bound. We make no adaptation claim here. The construction estimates $Q$ directly from a pseudo-outcome and never estimates $g$ or $p$.
Lemma (Integrated regression and contamination): The integrated regression $Q$ has a Hölder modulus of order $t_Q$, with a constant depending only on the fixed dimensions and radii. Let $\widetilde m:[0,1]^D\to[0,1]$ be any random function independent of a new block of $N$ trajectories. On that block regress the response $\widetilde m(X,L)$ among observations with $A_0=a_0$ on $X$ by a histogram of side length $h$, using $1/2$ in empty cells. Call the result $\widetilde Q$. Then, conditionally on $\widetilde m$, \[ \mathbb E[\|\widetilde Q-Q\|_{L^2(p)}^2\mid\widetilde m] \le C\left\{h^{2t_Q}+\frac{h^{-d_0}}{N+1} +\|\widetilde m-m\|_{L^2(\mu)}^2\right\}. \tag{7} \] No smoothness of the fitted function $\widetilde m$ is required.
Proof: For two covariate points, add and subtract $m(x',l)g(l\mid x)$: \[ Q(x)-Q(x')=\int[m(x,l)-m(x',l)]g(l\mid x)dl +\int m(x',l)[g(l\mid x)-g(l\mid x')]dl. \] The first term is bounded by a constant times $\|x-x'\|^{\min(s_m,1)}$, since $g$ integrates to one. The second is bounded by a constant times $\|x-x'\|^{\min(s_g,1)}$, since $0\le m\le1$ and the $l$-cube has volume one. Indices above one imply Lipschitz moduli through bounded first derivatives. The bounded support absorbs the constants needed to replace both exponents by $t_Q$.

For an $X$-cell $A$, write $\pi_A=\int_Ap$ and $\rho_A=\int_A e_0p\ge\varepsilon\pi_A$. The true selected mean of $Q$ is $q_A=\rho_A^{-1}\int_A e_0p Q$ and differs from every $Q(x)$ in that cell by at most $Ch^{t_Q}$. Conditional on $\widetilde m$, the selected pseudo-response mean in the cell is $q_A+r_A$, where \[ r_A=\frac1{\rho_A}\int_A e_0(x)p(x) \int [\widetilde m(x,l)-m(x,l)]g(l\mid x)\,dl\,dx. \] Jensen's inequality for the selected conditional law gives \[ \sum_A\pi_A r_A^2 \le\sum_A\frac{\pi_A}{\rho_A}\int_Ae_0p \int(\widetilde m-m)^2g\,dl\,dx \le\varepsilon^{-1}\|\widetilde m-m\|_{L^2(\mu)}^2. \tag{8} \] On a positive count $j$, the empirical pseudo-response mean is unbiased for $q_A+r_A$ and has conditional variance at most $1/(4j)$. The binomial reciprocal identity and empty-cell bound used earlier give an integrated sampling contribution at most $C h^{-d_0}/(N+1)$. Using $(u+v)^2\le2u^2+2v^2$ for approximation and contamination, then (8), proves (7). The argument conditioned on the whole random function first; it therefore does not assume that a fitted histogram is smooth or that its evaluation errors are independent across cells. ∎

Theorem (Constructive five-index upper bound): There is an explicitly tuned, five-block estimator with \[ \sup_{P\in\mathcal P(s_p,s_0,s_g,s_1,s_m)} \mathbb E_P(\widehat\psi_a-\psi_a)^2 \le C\left\{n^{-1}+n^{-2(\gamma_0+\gamma_Q)} +n^{-2(\gamma_0+\gamma_m)}+n^{-2(\gamma_1+\gamma_m)}\right\}. \tag{9} \] Its constant depends only on the fixed dimensions, indices, radii, density bounds and overlap. Thus \[ \gamma_0+\min(\gamma_Q,\gamma_m)\ge\tfrac12,\qquad \gamma_1+\gamma_m\ge\tfrac12 \tag{10} \] is an attained parametric-risk region. If the class contains a fixed nondegenerate constant Bernoulli outcome subfamily with uniform histories and constant assignments, its minimax risk is $\Theta(n^{-1})$ throughout (10).
Proof: For sufficiently large $n$, use five disjoint blocks of size $N=\lfloor n/5\rfloor$. On block 1 fit $h_0$ by regressing $I_0=\mathbf1\{A_0=a_0\}$ on $X$. On block 2 fit $h_1$ by regressing $I_1=\mathbf1\{A_1=a_1\}$ on $(X,L)$ among $I_0=1$. On block 3 fit $\widehat m$ by regressing $Y$ on $(X,L)$ among $I_0I_1=1$. Use equal-cube histograms with side lengths within a fixed factor of $N^{-1/(2t_0+d_0)}$, $N^{-1/(2t_1+D)}$, and $N^{-1/(2t_m+D)}$, respectively. Clip the propensity fits to $[\varepsilon,1-\varepsilon]$ and the outcome fit to $[0,1]$.

Here is the elementary risk calculation for these pilots. For each cell the regression oscillation is at most a constant times its side length to the relevant $t$ power. The positive-count response mean is unbiased for its selected cell mean and has variance at most $1/(4j)$. For the $h_1$ fit the selected cell probability is $\int_{\text{cell}}e_0\,d\mu\ge\varepsilon\mu(\text{cell})$; for the $\widehat m$ fit it is $\int_{\text{cell}}e_0e_1\,d\mu\ge\varepsilon^2\mu(\text{cell})$. For $h_0$ it is $P_X(\text{cell})$. Multiplying the binomial reciprocal and empty-cell bounds by the respective target cell masses and summing gives risk at most $C\{h^{2t}+h^{-d'}/(N+1)\}$ in dimension $d'$. This argument uses the intervention design measure $\mu=p g$ for the last two errors; it does not silently replace it by the observed marginal law of $(X,L)$. The resulting bounds are \[ \mathbb E\|h_0-e_0\|_{L^2(p)}^2\le Cn^{-2\gamma_0},\quad \mathbb E\|h_1-e_1\|_{L^2(\mu)}^2\le Cn^{-2\gamma_1},\quad \mathbb E\|\widehat m-m\|_{L^2(\mu)}^2\le Cn^{-2\gamma_m}. \tag{11} \]

On block 4 construct $\widehat Q$ from pseudo-response $\widehat m(X,L)$ as in (7), using side length $h_Q\asymp N^{-1/(2t_Q+d_0)}$. Independence of blocks 3 and 4, followed by (7) and (11), gives \[ \mathbb E\|\widehat Q-Q\|_{L^2(p)}^2 \le C\{n^{-2\gamma_Q}+n^{-2\gamma_m}\}. \tag{12} \] The fit already lies in $[0,1]$. On block 5 average the longitudinal score (1) and project its average to $[0,1]$.

The exact remainder (2) and bounded-score variance give (3). The function $h_0$ uses only block 1, whereas $\widehat Q$ uses only blocks 3 and 4. Thus their squared errors in the first product in (3) are independent. Similarly $h_1$ uses only block 2 and $\widehat m$ only block 3, so the second product also factors in expectation. Independence of $\widehat Q$ and $\widehat m$ is neither true nor needed. Substitution of (11)--(12) into (3) proves (9) using only second-moment pilot bounds, without the fourth-moment premise of (5). Small sample sizes are handled by the constant estimator and a larger constant. Condition (10) makes all displayed terms $O(n^{-1})$.

For the matching lower bound in that region, let histories be independent uniform variables, assignments independent Bernoulli$(1/2)$, and $Y\mid X,A_0,L,A_1\sim\operatorname{Ber}(\theta)$ with $\theta=1/2\pm b/\sqrt n$. The target is $\theta$ and the outcome law is constant in every continuous coordinate, hence lies in every stated smoothness ball under the explicit subfamily assumption. Product KL is $O(b^2)$. Choose fixed $b>0$ so total variation is at most $1/2$; the nearer-target test yields a lower bound $c/n$ for squared-error risk. Together with (9) this proves the restricted sharp rate. ∎

For a concrete improvement, take $d_0=d_1=1$, $s_0=s_1=s_m=1$ and $s_g=2/5$, with any positive $s_p$. Then \[ \gamma_0=1/3,\qquad \gamma_1=\gamma_m=1/4,\qquad \gamma_Q=2/9, \] so (10) holds. Separate histogram estimation of the transition density would instead yield \[ \gamma_g=\frac{2/5}{2(2/5)+2}=\frac17,\qquad \gamma_0+\min(\gamma_m,\gamma_g)=\frac{10}{21}<\frac12, \] which does not satisfy the earlier sufficient condition. The comparison concerns these proved first-order constructions; it is not a lower bound against every estimator that explicitly estimates $g$.

Unresolved minimax interactions

For fixed $T$, known assignment yields a parametric experiment, while for growing $T$ even completely smooth submodels require $n\varepsilon^T$ to diverge. The five-block construction now attains the displayed sufficient smoothness region with unknown propensities, without separately estimating the transition density. Its sufficient inequalities are not claimed to be necessary. Higher-order corrections, matching lower bounds in the slow regions, the full sharp five-index phase diagram and its fixed-multistage extension remain unresolved in this attempt.

Completion Estimate

35%

This is a heuristic estimate for the full catalogue target. The direct nested-regression theorem supplies actual pilot constructions, removes the earlier fourth-moment oracle premise for this bound and strictly enlarges a proved parametric region. A matching general lower bound across the slower five-index regions and the fixed-multistage characterization are still missing.

References

  • [agenda] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2025), arXiv:2508.17099v1. https://arxiv.org/html/2508.17099v1.
  • [robins] J. Robins, L. Li, E. Tchetgen Tchetgen and A. van der Vaart. Technical Report: Higher Order Influence Functions and Minimax Estimation of Nonlinear Functionals (2016). https://arxiv.org/abs/1601.05820.

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Econ-56-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-56

Minimax estimation of longitudinal intervention effects: A longitudinal remainder and an exact overlap subexperiment

Target

For a fixed two-stage regime $a=(a_0,a_1)$, write $e_0(x)=P(A_0=a_0\mid X=x)$, $e_1(x,l)=P(A_1=a_1\mid X=x,A_0=a_0,L=l)$, $g(l\mid x)=p(l\mid x,A_0=a_0)$, and $m(x,l)=\mathbb E[Y\mid x,a_0,l,a_1]$. Then \[ Q(x)=\int m(x,l)g(l\mid x)dl,\qquad \psi_a=\int Q(x)p(x)dx. \] The canonical question seeks the squared-error minimax rate under distinct Hölder restrictions on all five nuisance functions. The agenda is [agenda, Section 3.5.3]. Longitudinal higher-order methodology already exists, including [robins]; this note supplies an explicit first-order reduction and proves a sharp restricted experiment rather than asserting that the full longitudinal problem is solved.

Exact two-stage bias

Train pilots $h_0,h_1,\widehat m,\widehat Q$ independently of an evaluation block of size $N$, with $h_t\in[\varepsilon,1-\varepsilon]$ and outcome pilots in $[0,1]$. Put $I_t=\mathbf1\{A_t=a_t\}$ and average \[ S=\widehat Q(X)+\frac{I_0}{h_0(X)} \{\widehat m(X,L)-\widehat Q(X)\} +\frac{I_0I_1}{h_0(X)h_1(X,L)}\{Y-\widehat m(X,L)\}. \tag{1} \] Let $\mu(dx,dl)=p(x)g(l\mid x)dx\,dl$.
Theorem: Conditionally on the pilots, the bias of (1) is \[ B=\int p\left(1-\frac{e_0}{h_0}\right)(\widehat Q-Q)dx +\int\frac{e_0}{h_0}\left(1-\frac{e_1}{h_1}\right) (\widehat m-m)d\mu. \tag{2} \] Consequently its projected mean has risk bounded by \[ \frac{C_\varepsilon}{N}+C_\varepsilon\mathbb E\left[ \|h_0-e_0\|_{L^2(p)}^2\|\widehat Q-Q\|_{L^2(p)}^2 +\|h_1-e_1\|_{L^2(\mu)}^2\|\widehat m-m\|_{L^2(\mu)}^2\right]. \tag{3} \]
Proof: Conditional on $X=x$, the expectation of the second and third terms in (1) equals \[ \frac{e_0}{h_0}\int g\left\{\widehat m-\widehat Q +\frac{e_1}{h_1}(m-\widehat m)\right\}dl =\frac{e_0}{h_0}\left\{Q-\widehat Q+ \int g(1-e_1/h_1)(\widehat m-m)dl\right\}. \] Adding $\widehat Q$ proves (2). Cauchy--Schwarz and the clipped denominator bounds give the two products in (3), after $(x+y)^2\le2x^2+2y^2$. The score is bounded by $1+\varepsilon^{-1}+\varepsilon^{-2}$, so its conditional variance divided by $N$ proves the remaining term. ∎
Proposition: If $g\le G$, $\widehat g$ is a nonnegative conditional density, and $\widehat Q(x)=\int\widehat m(x,l)\widehat g(l\mid x)dl$, then \[ \|\widehat Q-Q\|_{L^2(p)} \le \|\widehat m-m\|_{L^2(\mu)} +\|\widehat g-g\|_{L^2(p(x)dx\,dl)}. \tag{4} \] In particular, suppose the evaluation size satisfies $N\asymp n$, and all four pilot errors have fourth moments of orders $n^{-4a_0},n^{-4a_1},n^{-4a_m},n^{-4a_g}$ in these respective norms. Then (3) is at most \[ C\{n^{-1}+n^{-2(a_0+\min(a_m,a_g))} +n^{-2(a_1+a_m)}\}. \tag{5} \]
Proof: Write $\widehat Q-Q=\int(\widehat m-m)g+ \int\widehat m(\widehat g-g)$. Jensen bounds the first term's norm by the first term on the right of (4). Cauchy--Schwarz in $l$, whose unit cube has volume one, and $0\le\widehat m\le1$ bound the second term. Minkowski proves (4). For (5), apply Cauchy--Schwarz to the expectations of products in (3), then (4) and $(x+y)^4\le8(x^4+y^4)$. ∎
Thus an explicitly sufficient parametric region for any pilots satisfying these moment bounds is $a_0+\min(a_m,a_g)\ge1/2$ and $a_1+a_m\ge1/2$. This is a conditional statement about pilot performance, not an unproved assertion that every indicated exponent is attainable for arbitrary joint Hölder indices. Formula (2) also shows that density and transition errors do not necessarily enter as independent additive first-order errors. In particular, a sufficiently good direct $Q$ estimator may be preferable to separately estimating $g$.

An exact result with supplied assignment probabilities

Consider $T$ treatment decisions, each with the probability of the requested action at least $\varepsilon$. Histories and their transitions may otherwise be arbitrary. Let $p_t$ denote those supplied, history-dependent probabilities, $Y\in[0,1]$, and retain sequential exchangeability and consistency.
Theorem: The minimax risk over this known-assignment experiment is at most \[ \min\{1,\,(n\varepsilon^T)^{-1}\}. \] Over the subexperiment with independent assignments of probability $\varepsilon$, ancillary uniform continuous histories, and an unknown constant Bernoulli outcome mean only on the requested treatment path, it is at least $c\min\{1,(n\varepsilon^T)^{-1}\}$ for a universal $c>0$. All continuous-coordinate transition and outcome functions in this subexperiment are smooth of every order.
Proof: Let $R=\prod_{t=1}^T\mathbf1\{A_t=a_t\}/p_t(H_t)$. Iterated conditional expectations, or cancellation of the assignment factors in the joint density, give $\mathbb E[RY]=\psi_a$ and $\mathbb E R=1$. Since $R\le\varepsilon^{-T}$, $\mathbb E(RY)^2\le\mathbb E R^2\le\varepsilon^{-T}$. The projected empirical mean therefore has risk at most $(n\varepsilon^T)^{-1}$; the constant estimator gives the other bound. For the lower bound let $q=\varepsilon^T$, set all off-path outcome means to $1/2$, and set the on-path mean to $1/2\pm\Delta$. The one-observation KL is at most $Cq\Delta^2$ for $0<\Delta\le1/4$. Choose $\Delta=b\min\{1,(nq)^{-1/2}\}$ with $b$ small. Then sample total variation is at most $1/2$ and the two targets differ by $2\Delta$. Testing by the nearer target gives maximal squared risk at least $\Delta^2/4$. Uniform ancillary histories do not change this likelihood calculation or any smoothness constraint with sufficient fixed radius. ∎

Unresolved minimax interactions

For fixed $T$, known assignment yields a parametric experiment, while for growing $T$ even completely smooth submodels require $n\varepsilon^T$ to diverge. Neither fact establishes the sharp rate with unknown propensities. Equations (2)--(5) isolate the two first-order bottlenecks; they leave open higher-order corrections, the exact smoothness of the integrated regression $Q$, and matching lower bounds coupling $e_0,e_1,g,m$. The full five-index rate and its fixed-multistage extension remain unresolved.

References

  • [agenda] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2025), arXiv:2508.17099v1. https://arxiv.org/html/2508.17099v1.
  • [robins] J. Robins, L. Li, E. Tchetgen Tchetgen and A. van der Vaart. Technical Report: Higher Order Influence Functions and Minimax Estimation of Nonlinear Functionals (2016). https://arxiv.org/abs/1601.05820.

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